Introduction

• Signals can be analyzed in both time domain and frequency domain.
• The frequency content of a signal describes how different frequency components contribute to the overall signal.
• Applications in biomedical signals, audio processing, communications, and image processing.

Convolution in Time Domain

• Convolution is a fundamental operation in signal processing.

• Given two signals ( x(t) ) and ( h(t) ), their convolution is defined as:

  \[ y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau \]

• In discrete-time, convolution is:

  \[ y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \]

Convolution Theorem

• Convolution in time domain corresponds to multiplication in frequency domain:

  \[ X(f) H(f) = Y(f) \]

• This property is crucial in filter design and system analysis.

Introduction to Fourier Series

(-1.0, 4.0)

(-1.0, 4.0)

Introduction to Fourier Series

• Fourier series represents periodic signals as a sum of sinusoids:

  \[ x(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t} \]

  where ( C_n ) are the Fourier coefficients.

• Decomposing a signal into sinusoidal components allows frequency analysis.

Fourier Coefficients

• The Fourier coefficients ( C_n ) are computed as:

  \[ C_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-jn\omega_0 t} dt \]

• Determines how much of each frequency is present in the signal.

Example of Fourier Series Expansion

Relationship Between Frequency Content and Sampling Frequency

• Sampling theorem: A signal must be sampled at a frequency at least twice its highest frequency component:

  \[ f_s \geq 2 f_{max} \]

• Aliasing occurs if sampling frequency is too low.